Non-over-1 temperament: Difference between revisions
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A | A '''non-over-1 temperament''' is a [[regular temperament]] that tempers a [[subgroup]] corresponding to a harmonic series chord r:n<sub>1</sub>:...:n<sub>k</sub> where r ≠ 1, but is not meant to approximate a chord of the form 1:m<sub>1</sub>:...:m<sub>k</sub>. Non-over-1 temperaments give regular-temperament interpretations to edos that approximate [[Overtone scales|over-1]] chords such as 4:5:6:7:11 poorly, such as [[14edo]], [[18edo]], [[23edo]] and [[29edo]]. | ||
== Examples == | == Examples == | ||
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Tridec is a temperament generated by a generator near 455.2178¢ (for example, [[29edo|11\29]] or [[37edo|14\37]]) and has an 8-note MOS. If you restrict to the 8-note MOS, Tridec is an 2.7/5.11/5.13/5 temperament that tempers the chord 5:7:11:13; one generator represents a 13/10, three generators represent a 11/10, -4 generators represent a 7/5. | Tridec is a temperament generated by a generator near 455.2178¢ (for example, [[29edo|11\29]] or [[37edo|14\37]]) and has an 8-note MOS. If you restrict to the 8-note MOS, Tridec is an 2.7/5.11/5.13/5 temperament that tempers the chord 5:7:11:13; one generator represents a 13/10, three generators represent a 11/10, -4 generators represent a 7/5. | ||
Tridec essentially contains all the notes of 2.3.5 [[porcupine]] temperament and satisfies all its relations; hence it is essentially the same as 13-limit [[Ammonite]]; however, provided you restrict yourself to the 8-note MOS, you're using it as a non-over- | Tridec essentially contains all the notes of 2.3.5 [[porcupine]] temperament and satisfies all its relations; hence it is essentially the same as 13-limit [[Ammonite]]; however, provided you restrict yourself to the 8-note MOS, you're using it as a non-over-1 temperament. | ||
=== Sensi === | === Sensi === | ||
[[Sensi]] is ''effectively'' a non-over- | [[Sensi]] is ''effectively'' a non-over-1 temperament provided you restrict yourself to the sensi[8] MOS. The sensi[8] MOS only has a 5:6:7:9:13 chord, but no chord of the form 2:m<sub>1</sub>:...:m<sub>k</sub> (except 2:3). Thus sensi can be viewed as a 2.6/5.7/5.9/5.13/10 or 2.3.6/5.7/5.13/10 temperament. (See [http://x31eq.com/cgi-bin/rt.cgi?limit=2_6%2F5_7%2F5_9%2F5_13%2F10&ets=19_27&tuning=po&subgroup=on x31eq data page].) | ||
{| class="wikitable" | {| class="wikitable" | ||
Revision as of 03:49, 8 February 2021
A non-over-1 temperament is a regular temperament that tempers a subgroup corresponding to a harmonic series chord r:n1:...:nk where r ≠ 1, but is not meant to approximate a chord of the form 1:m1:...:mk. Non-over-1 temperaments give regular-temperament interpretations to edos that approximate over-1 chords such as 4:5:6:7:11 poorly, such as 14edo, 18edo, 23edo and 29edo.
Examples
Tridec
Tridec is a temperament generated by a generator near 455.2178¢ (for example, 11\29 or 14\37) and has an 8-note MOS. If you restrict to the 8-note MOS, Tridec is an 2.7/5.11/5.13/5 temperament that tempers the chord 5:7:11:13; one generator represents a 13/10, three generators represent a 11/10, -4 generators represent a 7/5.
Tridec essentially contains all the notes of 2.3.5 porcupine temperament and satisfies all its relations; hence it is essentially the same as 13-limit Ammonite; however, provided you restrict yourself to the 8-note MOS, you're using it as a non-over-1 temperament.
Sensi
Sensi is effectively a non-over-1 temperament provided you restrict yourself to the sensi[8] MOS. The sensi[8] MOS only has a 5:6:7:9:13 chord, but no chord of the form 2:m1:...:mk (except 2:3). Thus sensi can be viewed as a 2.6/5.7/5.9/5.13/10 or 2.3.6/5.7/5.13/10 temperament. (See x31eq data page.)
| Generators | Cents* | Approximate ratios† |
|---|---|---|
| 0 | 0.000 | 1/1 |
| 1 | 443.322 | 13/10~9/7 |
| 2 | 886.644 | 42/25~5/3 |
| 3 | 129.966 | 13/12~14/13~15/14~27/25 |
| 4 | 573.288 | 7/5~25/18~18/13 |
| 5 | 1016.610 | 9/5~70/39 |
| 6 | 259.932 | 7/6~15/13 |
| 7 | 703.253 | 3/2 |
- * in 2.3.5.7.13 POTE tuning
- † 2.3.5.7.13 ratio interpretations